# Controllability of Fractional Functional Evolution Equations of Sobolev Type via Characteristic Solution Operators

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DOI: 10.1007/s10957-012-0174-7

- Cite this article as:
- Fec̆kan, M., Wang, J. & Zhou, Y. J Optim Theory Appl (2013) 156: 79. doi:10.1007/s10957-012-0174-7

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## Abstract

The paper is concerned with the controllability of fractional functional evolution equations of Sobolev type in Banach spaces. With the help of two new characteristic solution operators and their properties, such as boundedness and compactness, we present the controllability results corresponding to two admissible control sets via the well-known Schauder fixed point theorem. Finally, an example is given to illustrate our theoretical results.

### Keywords

ControllabilityFractional derivativeFunctional evolution equationsSobolevCharacteristic solution operators## 1 Introduction

A strong motivation for studying fractional evolution equations comes from the fact they have been proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics, and economics. For more details on the theory and application in this field, one can see the monographs of Diethelm [1], Kilbas et al. [2], Miller and Ross [3], Podlubny [4] and Tarasov [5].

Recently, the existence of mild solutions, Mittag–Leffer–Ulam stability, controllability and optimal controls for all kinds of fractional semilinear evolution system in Banach spaces have been reported by many researchers. We refer the reader to El-Borai [6, 7], Balachandran and Park [8], Zhou and Jiao [9, 10], Hernández et al. [11], Wang and Zhou [12–14], Sakthivel et al. [15], Debbouchea and Baleanu [16], Wang et al. [17–23], Kumar and Sukavanam [24], Li et al. [25] and the references therein.

It is remarkable that Balachandra and Dauer [26] provide some sufficient conditions for controllability of integer functional evolution equations of Sobolev type by virtue of the theory of semigroup theory via the techniques of fixed point theorem. Very recently, Li et al. [27] obtain the existence results for fractional evolution equations of Sobolev type by virtue of the theory of propagation family via the techniques of the measure of noncompactness and the condensing maps. However, controllability of fractional functional evolution equations of Sobolev type have not been extensively via the theory of semigroup theory or propagation family.

Motivated by [9, 26, 27], we offer to study the controllability of a class of fractional functional evolution equations of Sobolev type via the theory of semigroup theory. Our aim in this paper is to provide some suitable sufficient conditions for the controllability results corresponding to two admissible control sets without assuming the semigroup is compact. To achieve our purpose, we have to introduce two characteristic solution operators and study their properties such as boundedness and compactness. To prove the controllability results we follow the techniques similar to that of [26] with some necessary modifications so as to be compatible with fractional functional evolution equations. But we emphasize that the assumptions on the closed operator for *E* (see [*C*_{1}] in [26]) and the compact resolvent operator (see [*C*_{4}] in [26]) does not need. Here, we will remove these redundant conditions to obtain the controllability results.

## 2 Preliminaries

*X*and

*Y*be two real Banach spaces. We consider the following fractional functional evolution equations of Sobolev type:

*q*<1 with the lower limit zero, the operators

*A*:

*D*(

*A*)⊂

*X*→

*Y*and

*E*:

*D*(

*E*)⊂

*X*→

*Y*, the state

*x*(⋅) takes values in

*X*and the control function

*u*(⋅) is given in , the Banach space of admissible control functions with

*U*a Banach space where either for \(\frac{1}{2}<q<1\) or for 0<

*q*<1. Moreover,

*B*is a bounded linear operator from

*U*into

*Y*.

*f*:

*J*×

*C*→

*Y*with

*C*:=

*C*([−

*r*,0],

*X*) will be specified later.

*x*:

*J*

^{∗}:=[−

*r*,

*a*]→

*X*is continuous,

*x*

_{t}is the element of

*C*defined by

*x*

_{t}(

*s*):=

*x*(

*t*+

*s*),−

*r*≤

*s*≤0. The domain

*D*(

*E*) of

*E*becomes a Banach space with norm ∥

*x*∥

_{D(E)}:=∥

*Ex*∥

_{Y},

*x*∈

*D*(

*E*) and

*ϕ*∈

*C*(

*E*):=

*C*([−

*r*,0],

*D*(

*E*)).

Let us first recall the following known definitions. For more details, see [2].

### Definition 2.1

*γ*with the lower limit zero for a function

*f*:[0,∞[→ℝ is defined as

*Γ*(⋅) is the Gamma function, which is defined by \(\varGamma(\gamma):=\int_{0}^{\infty}t^{\gamma-1}e^{-t}\,dt\).

### Definition 2.2

*γ*with the lower limit zero for a function

*f*:[0,∞[→ℝ can be written as

### Definition 2.3

*γ*for a function

*f*:[0,∞[→ℝ can be written as

## 3 Characteristic Solution Operators and their Properties

*A*and

*E*.

- (H
_{1}): *A*and*E*are linear operators, and*A*is closed.- (H
_{2}): *D*(*E*)⊂*D*(*A*) and*E*is bijective.- (H
_{3}): Linear operator

*E*^{−1}:*Y*→*D*(*E*)⊂*X*is compact (which implies that*E*^{−1}is bounded).

Note (H_{3}) implies that *E* is closed since the fact: *E*^{−1} is closed and injective, then its inverse is also closed. It comes from (H_{1})–(H_{3}) and the closed graph theorem, we obtain the boundedness of the linear operator −*AE*^{−1}:*Y*→*Y*. Consequently, −*AE*^{−1} generates a semigroup {*T*(*t*),*t*≥0}, \(T(t):=e^{-AE^{-1}t}\). We suppose *M*_{1}:=sup_{t≥0}∥*T*(*t*)∥<∞.

### Lemma 3.1

*If the formula*(3)

*holds*,

*then we have*

*Here*

*and*

*are called characteristic solution operators and given by*

*where*

*ξ*

_{q}

*is a probability density function defined on*]0,∞[.

### Proof

*λ*>0. Formally applying the following Laplace transforms: to the first equation in (3), we obtain which implies that where

*I*is the identity operator defined on

*Y*. Thus,

### Definition 3.1

*ϕ*∈

*C*(

*E*), a mild solution of the system (1) we mean a function

*x*∈

*C*(

*J*

^{∗},

*X*) which satisfies

The following results of and will be used throughout this paper.

### Lemma 3.2

### Proof

(i) For any fixed *t*≥0, since *E*^{−1} and *T*(*t*) are linear operator, it is easy to see that and are also linear operators.

*x*∈

*X*, according to (4), we have where we use the fact

*k*, set

*Y*

_{k}:={

*x*∈

*Y*:∥

*x*∥≤

*k*}. Then

*Y*

_{k}is clearly a bounded subset in

*Y*. We need prove that for any positive constant

*k*and

*t*≥0, the following two sets: are relatively compact in

*Y*. By the assertion (i), we know and are also linear and bounded. So they maps

*Y*

_{k}into a bounded subsets of

*Y*. Then and are relatively compact in

*Y*for any

*k*>0 and

*t*≥0 due to the compactness of

*E*

^{−1}:

*Y*→

*X*. The proof is complete. □

## 4 Main Results

In this section, we study the controllability of the system (1) via the well-known Schauder fixed point theorem.

### Definition 4.1

The fractional system (1) is said to be controllable on the interval *J* iff for every continuous initial function *ϕ*∈*C*(*E*) and every *x*_{1}∈*D*(*E*) there exists a control such that the mild solution *x* of system (1) satisfies *x*(*a*)=*x*_{1}.

_{1})–(H

_{3}), we need the following assumptions:

*Wu*∈

*D*(

*E*) and

*W*is well defined. In fact, it holds for \(q\in ]\frac{1}{2},1[\) and

*u*∈

*L*

^{2}(

*J*,

*U*), meanwhile, for

*q*∈]0,1[ and

*u*∈

*L*

^{∞}(

*J*,

*U*).

*t*∈

*J*, where \(K_{q}:=\sqrt{\frac{a^{2q-1}}{2q-1}}\) for \(q\in ]\frac{1}{2},1[\) and

*u*∈

*L*

^{2}(

*J*,

*U*), meanwhile, \(K_{q}:=\frac{a^{q}}{q}\) for

*q*∈]0,1[ and

*u*∈

*L*

^{∞}(

*J*,

*U*).

- (H
_{5}):

*x*(⋅), it is suitable to define the following control formula:

*u*in (8), the operator \(\mathcal {P}\) defined by from

*C*(

*J*

^{∗},

*X*) into

*C*(

*J*

^{∗},

*X*) for each

*x*∈

*C*(

*J*

^{∗},

*X*), has a fixed point. Clearly, this fixed point is just a solution of system (1). Further, one can check which means that

*u*steers the fractional system (1) from

*ϕ*(0) to

*x*

_{1}in finite time

*a*. Consequently, we claim the system (1) is controllable on

*J*.

For each number *k*>0, define \(\mathcal {B}_{k}:=\{x\in C(J^{*},X): \|x(t)\|\leq k,~t\in J^{*}\}\). Of course, \(\mathcal {B}_{k}\) is clearly a bounded, closed, convex subset in *C*(*J*^{∗},*X*).

Under the assumptions (H_{1})–(H_{5}), we will establish three important results as follows.

### Lemma 4.1

### Proof

*t*∈[−

*r*,0] then \(\|(\mathcal {P}x)(t)\|=\|\phi(t)\|\le \max_{t\in[-r,0]}\|\phi(t)\|\). If

*t*∈[0,

*a*] then we derive where we note that the control

*u*defined in (8) satisfies which implies that Thus \(\mathcal {P}\mathcal {B}_{K}\subset \mathcal {B}_{K}\) for any \(K\geq \max\{\max_{t\in[-r,0]}\|\phi(t)\|,\frac{M^{*}}{1-\rho}\}\) sufficiently large. The proof is complete. □

### Lemma 4.2

*For every fixed**t*∈*J**the set*\(V_{K}(t):=\{(\mathcal {P}x)(t): x\in \mathcal {B}_{K}\} \)*is precompact in**X*.

### Proof

*t*∈[−

*r*,0], since

*V*

_{K}(

*t*)={

*ϕ*(

*t*)}. So let 0<

*t*<

*a*be fixed. Note Next, for \(x\in \mathcal {B}_{K}\), we derive and so \(\{(\mathcal {P}_{0}x)(t): x\in \mathcal {B}_{K}\}\) is bounded in

*Y*by (10).

Since *E*^{−1}:*Y*→*X* is compact, then \((\mathcal {P}x)(t):=E^{-1}\left(\{(\mathcal {P}_{0}x)(t): x\in \mathcal {B}_{K}\}\right)\) is precompact in *X*. The proof is complete. □

### Lemma 4.3

\(\mathcal {P}\mathcal {B}_{K}:= \{\mathcal {P}x : x\in \mathcal {B}_{K}\}\)*is equicontinuous*.

### Proof

Note that Lemma 3.2(ii), and are continuous in the uniform operator topology for *t*≥0, and sup_{s∈J}|*g*_{K}(*s*)|<∞ and *u*(⋅) is bounded by (10). We can obtain the terms *I*_{1},*I*_{3},*I*_{5},*I*_{6},*I*_{7}→0 as *t*″→*t*′. Moreover, applying Lebesgue’s dominated convergence theorem, one can check the terms *I*_{2},*I*_{4}→0 as *t*″→*t*′. Thus, \(\mathcal {P}\mathcal {B}_{K}\) is equicontinuous and also bounded. □

Now we are ready to state the main result in this paper.

### Theorem 4.1

*Assume* (H_{1})*–*(H_{5}) *are satisfied*. *Then the system* (1) *is controllable on**J**provided that the condition* (9) *hold*.

### Proof

It follows Lemmas 4.1–4.3 and the Arzela–Ascoli theorem that \(\mathcal {P}\mathcal {B}_{K}\) is precompact in *C*(*J*^{∗},*X*). Hence \(\mathcal {P}\) is a completely continuous operator on *C*(*J*^{∗},*X*). From the Schauder fixed point theorem, \(\mathcal {P}\) has a fixed point in \(\mathcal {B}_{K}\). Any fixed point of \(\mathcal {P}\) is a mild solution of the system (1) on *J* satisfying \((\mathcal {P}x)(t)= x(t)\in X\). Thus, the system (1) is controllable on *J*. □

## 5 An Example

*X*=

*Y*=

*U*:=

*L*

^{2}[0,

*π*]. Consider the following fractional partial differential equation with control:

*A*:

*D*(

*A*)⊂

*X*→

*Y*by

*Ax*:=−

*x*

_{yy}and

*E*:

*D*(

*E*)⊂

*X*→

*Y*by

*Ex*:=

*x*−

*x*

_{yy}, where each domain

*D*(

*A*),

*D*(

*E*) is given by {

*x*∈

*X*:

*x*,

*x*

_{y}are absolutely continuous,

*x*

_{yy}∈

*X*,

*x*(

*t*,0)=

*x*(

*t*,

*π*)=0}. From [28],

*A*and

*E*can be written as respectively, where \(x_{n}(y):= \sqrt{\frac{2}{\pi}}\sin ny,n = 1,2,\ldots\) is the orthonormal set of eigenfunctions of

*A*. Moreover, for any

*x*∈

*X*we have

*E*

^{−1}is compact, bounded with ∥

*E*

^{−1}∥≤1 and −

*AE*

^{−1}generates the above strongly continuous semigroup

*T*(

*t*) on

*Y*with ∥

*T*(

*t*)∥≤

*e*

^{−t}≤1. Then, the two characterize operators and can be written as Clearly,

- (C
_{1}): *B*:*U*→*Y*is defined by*B*:=*bI*,*b*>0 and is defined by Since \(q=\frac{3}{4}>\frac{1}{2}\), we take and so \(K_{\frac{3}{4}}:=\sqrt{2}\). Next, let*u*(*s*,*y*):=*x*(*y*)∈*U*. Then whereis a Mittag–Leffler function (for more details, see formulas (24)–(27) of [29]).$$\mathbb{E}_{\frac{3}{4}}\left(-\frac{n^{2}}{1+n^{2}}\right):=\int_{0}^{\infty} e^{-\frac{n^{2}}{1+n^{2}}\theta}\xi_{\frac{3}{4}}(\theta)\,d\theta= \int_{0}^{\infty} e^{-\frac{n^{2}}{1+n^{2}}\theta}\frac{1}{\frac{3}{4}}\theta^{-\big(1+\frac{1}{\frac{3}{4}}\big)}\varpi_{\frac{3}{4}} (\theta^{-\frac{1}{\frac{3}{4}}})\,d\theta, $$

*θ*>0. So we have

*W*is surjective. So we define an inverse by

*x*∈

*D*(

*E*), we derive Note

*W*

^{−1}

*x*is independent of

*t*∈

*J*

_{1}. Consequently, we obtain

- (C
_{2}): *f*:*J*_{1}×*R*→*R*. For each*x*∈*R*,*f*(*t*,*x*) is measurable and for each*t*∈*J*_{1},*f*(*t*,*x*) is continuous. Moreover, \(\limsup_{k\rightarrow \infty}\frac{1}{k}\sup_{t\in J_{1},|x|\leq k}|f(t,x)|:= \gamma<\infty\).

*F*:

*J*

_{1}×

*C*([−1,0],

*X*)→

*Y*by

*F*(

*t*,

*z*)(

*y*)=

*f*(

*t*,

*z*(−

*r*)(

*y*)). Now, the system (11) can be abstracted as

*J*

_{1}.

*h*has a unique stationary point \(s^{*}=(\frac{\sqrt{2}}{2})^{\frac{4}{3}}\) and

*h*″(

*s*)>0 for

*s*≥0 that \(h(s)\ge h(s^{*})=\frac{1}{2}\) for

*s*≥0. Thus, we obtain

*η*≥30. On the other hand, a numerical computation in Mathematica shows

*γ*<0.17508.

## 6 Conclusions

Controllability of fractional functional evolution equations of Sobolev type have been investigated. Utilizing the boundedness and compactness of two new introduced characteristic solution operators with Schauder fixed point theorem, sufficient conditions for controllability of such case of equations corresponding to two certain admissible control sets are provided. Here, we succeed in removing the compactness condition on the semigroup.

Our further work will be devoted to study controllability of the above problems with the help of the theory of propagation family and the techniques of the measure of noncompactness.

## Acknowledgements

The authors thank the referees for their careful reading of the manuscript and insightful comments. We also acknowledge the valuable comments and suggestions from the editors. Finally, The first author acknowledges the support by Grants VEGA-MS 1/0507/11, VEGA-SAV 2/0124/12 and APVV-0414-07; the second author acknowledges the support by National Natural Science Foundation of China (11201091) and Key Projects of Science and Technology Research in the Chinese Ministry of Education (211169) and the third author acknowledges the support by National Natural Science Foundation of China (11271309), Specialized Research Fund for the Doctoral Program of Higher Education (20114301110001) and Key Projects of Hunan Provincial Natural Science Foundation of China (12JJ2001).